Percentages, Ratios and Proportions | 百分比、比率和比例

📚 Percentages, Ratios and Proportions | 百分比、比率和比例

Percentages, ratios and proportions are fundamental tools in mathematics that help us compare quantities, understand relationships, and solve real-world problems. Whether you are calculating discounts in a shop, mixing ingredients in a recipe, or scaling up a map, these concepts allow you to work with parts of a whole and relative sizes with confidence. In this article, we will explore key definitions, conversion methods, and problem-solving strategies to build a solid understanding of these interconnected topics, as covered in the Cambridge KS3 curriculum.

百分比、比率和比例是数学中的基本工具，帮助我们比较数量、理解关系并解决现实世界的问题。无论你是在商店里计算折扣、混合食谱中的配料，还是放大地图比例，这些概念都能让你自信地处理整体中的部分和相对大小。在本文中，我们将探讨关键定义、转换方法和解决问题的策略，以建立对这些相互关联主题的扎实理解，涵盖剑桥KS3课程的内容。

1. Understanding Percentages | 理解百分比

A percentage is a way of expressing a number as a fraction of 100. The word ‘percent’ literally means ‘per hundred’, so 45% means 45 out of every 100. Percentages are used widely in everyday life, from test scores to financial interest rates, because they provide a standard scale for comparison.

百分比是一种将数字表示为百分之几的方式。“百分之”一词的字面意思是“每百”，因此45%意味着每100份中的45份。百分比在日常生活中被广泛使用，从考试分数到金融利率，因为它们提供了一个标准化的比较尺度。

To convert a percentage to a fraction, write it over 100 and simplify if possible. For example, 65% = 65/100 = 13/20. To convert a fraction to a percentage, find an equivalent fraction with a denominator of 100, or multiply the fraction by 100%. For instance, 3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%. When converting between decimals and percentages, remember that a decimal can be multiplied by 100 to get the percentage, and a percentage can be divided by 100 to get the decimal. So 0.82 = 82%, and 7% = 0.07.

要将百分比转换为分数，将其写成分母为100的分数并尽可能简化。例如，65% = 65/100 = 13/20。要将分数转换为百分比，找出一个分母为100的等值分数，或者将该分数乘以100%。例如，3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%。在十进制与百分比之间转换时，记住十进制数乘以100得到百分比，百分比除以100得到十进制数。因此0.82 = 82%，而7% = 0.07。

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2. Calculating Percentage of an Amount | 计算一个数量的百分比

Finding a percentage of a quantity is a common task. There are two main methods: the decimal method and the fraction method. Using the decimal method, you convert the percentage to a decimal and multiply. For example, to find 30% of 240, calculate 0.30 × 240 = 72. With the fraction method, you write the percentage as a fraction over 100 and then multiply. So 30% of 240 becomes (30/100) × 240 = (30 × 240)/100 = 72.

求一个数量的百分比是一项常见任务。主要有两种方法：十进制法和分数法。使用十进制法，将百分比转换为十进制数然后相乘。例如，求240的30%，计算0.30 × 240 = 72。使用分数法，将百分比写成分母为100的分数，然后相乘。因此240的30% 变为 (30/100) × 240 = (30 × 240)/100 = 72。

For mental calculations, it helps to use benchmark percentages like 10%, 5% and 1%. To find 10%, divide by 10; to find 5%, find 10% and halve it; to find 1%, divide by 100. These can then be combined. For instance, to find 17% of a number, you could find 10% + 5% + (2 × 1%).

对于心算，使用基准百分比如10%、5%和1%会很有帮助。求10%，除以10；求5%，先求10%然后减半；求1%，除以100。这些可以组合使用。例如，求一个数的17%，可以先求10% + 5% + (2 × 1%)。

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3. Percentage Increase and Decrease | 百分比增减

Percentage change is used to describe how much a quantity has grown or shrunk relative to its original value. A percentage increase occurs when a value goes up, such as a price rise of 15%. A percentage decrease happens when a value goes down, such as a 20% discount in a sale.

百分比变化用于描述一个数量相对于其原始值增长或缩小了多少。当数值上升时，例如价格上涨15%，就会发生百分比增加。当数值下降时，例如销售中打折20%，就会发生百分比减少。

To calculate a percentage increase, first find the increase amount by multiplying the original value by the percentage (as a decimal), then add it to the original. Alternatively, you can use a multiplier. For an increase of 15%, the multiplier is 1 + 0.15 = 1.15. So a £80 coat after a 15% increase becomes £80 × 1.15 = £92. For a percentage decrease, subtract the percentage from 100% to find the multiplier. A 20% decrease means you pay 80%, so the multiplier is 0.80. The same coat with a 20% discount costs £80 × 0.80 = £64.

计算百分比增加时，首先将原始值乘以百分比（以十进制表示）得到增加量，然后将其加到原始值上。或者，你可以使用乘数。对于增加15%，乘数为 1 + 0.15 = 1.15。因此，一件80英镑的外套增加15%后变为 80 × 1.15 = 92 英镑。对于百分比减少，从100%中减去百分比以找到乘数。减少20%意味着你只需支付80%，因此乘数为0.80。同一件外套享受20%折扣后价格为 80 × 0.80 = 64 英镑。

To find the original value after a percentage change, you reverse the process by dividing by the multiplier. If a price including 20% VAT is £96, the original price before VAT was £96 ÷ 1.20 = £80.

要在百分比变化后求原始值，可以通过除以乘数来逆转过程。如果包含20%增值税的价格为96英镑，则增值税前的原价为 96 ÷ 1.20 = 80 英镑。

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4. Introduction to Ratios | 比率入门

A ratio compares two or more quantities, showing the relative size of one quantity to another. Ratios can be written in several forms: using a colon, as in 3:2; as a fraction, 3/2; or with the word ‘to’, 3 to 2. In a ratio, the order of the numbers is important. The ratio 3:2 is not the same as 2:3.

比率比较两个或多个数量，显示一个数量相对于另一个数量的大小。比率可以用几种形式书写：使用冒号，如 3:2；作为分数，3/2；或用“比”字，3比2。在比率中，数字的顺序很重要。比率 3:2 与 2:3 不相同。

Ratios can be simplified just like fractions by dividing all parts by a common factor. For example, the ratio 10:15 can be simplified by dividing both numbers by 5, giving 2:3. A ratio in its simplest form has no common factor other than 1.

比率可以像分数一样通过将所有部分除以一个公因数来简化。例如，比率 10:15 可以通过将两个数字都除以5来简化，得到 2:3。最简形式的比率除了1之外没有公因数。

When a ratio involves three or more quantities, the same simplification rules apply. For instance, 24:36:60 divided by 12 gives 2:3:5.

当比率涉及三个或更多数量时，相同的简化规则适用。例如，24:36:60 除以12得到 2:3:5。

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5. Sharing in a Given Ratio | 按给定比率分配

One of the most practical applications of ratios is dividing an amount into parts according to a specific ratio. The key idea is to work out the total number of parts, then find the value of one part, and finally multiply to find each share.

比率最实用的应用之一是按照特定比率将一个总量划分为若干部分。关键思想是计算出总份数，然后求出一份的值，最后相乘求出每一份。

For example, if £60 is shared between two people in the ratio 3:2, the total number of parts is 3 + 2 = 5 parts. One part is £60 ÷ 5 = £12. The first person gets 3 parts: 3 × £12 = £36, and the second person gets 2 parts: 2 × £12 = £24.

例如，如果 60 英镑按照比率 3:2 分给两个人，总份数为 3 + 2 = 5 份。一份为 60 ÷ 5 = 12 英镑。第一个人得到3份：3 × 12 = 36 英镑，第二个人得到2份：2 × 12 = 24 英镑。

This method also works for three or more shares. Divide £180 in the ratio 1:2:3: total parts = 1+2+3 = 6, one part = £30, so shares are £30, £60 and £90.

此方法也适用于三份或更多份的划分。将 180 英镑按照比率 1:2:3 分配：总份数 = 1+2+3 = 6，一份为 30 英镑，因此三份分别为 30 英镑、60 英镑和 90 英镑。

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6. Ratios and Scales | 比率与比例尺

Ratios are used extensively in maps, scale drawings, and models. A scale ratio such as 1:50 000 means that 1 unit on the map represents 50 000 units in real life. If a map has a scale of 1:25 000, then 1 cm on the map equals 25 000 cm in reality, which is 250 m.

比率广泛用于地图、比例图和模型中。比例尺比率如 1:50,000 表示地图上的 1 个单位代表现实中的 50,000 个单位。如果地图的比例尺为 1:25,000，那么地图上的 1 厘米等于现实中的 25,000 厘米，即 250 米。

To find the actual length from a scale drawing, multiply the drawing measurement by the scale factor. Conversely, to find the drawing measurement from an actual length, divide by the scale factor. Always ensure the units are consistent.

要从比例图求实际长度，将图上的测量值乘以比例因子。相反，从实际长度求图上的测量值，则除以比例因子。务必确保单位一致。

Scale	Map distance	Real distance
1:100 000	5 cm	5 km
1:50	6 cm	3 m

When the scale is written as a ratio without units, it is because the same unit is used on both sides. This makes it easy to convert between any units.

当比例尺写成一个没有单位的比率时，是因为两边使用相同的单位。这使得在任何单位之间转换都变得容易。

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7. Understanding Proportion | 理解比例

Proportion tells us how two quantities are related. If two quantities are in direct proportion, then as one increases, the other increases at the same rate. For example, if 3 apples cost 90p, then 6 apples will cost 180p, because the cost is directly proportional to the number of apples. The ratio between cost and number is constant.

比例告诉我们两个量是如何关联的。如果两个量成正比，那么当一个量增加时，另一个也以相同的速率增加。例如，如果 3 个苹果花费 90 便士，那么 6 个苹果将花费 180 便士，因为成本与苹果的数量成正比。成本与数量之间的比率是恒定的。

To solve direct proportion problems, you can use the unitary method: first find the value of one unit, then multiply to find the required value. For 3 apples costing 90p, one apple costs 90 ÷ 3 = 30p. Then 7 apples cost 7 × 30p = 210p.

解决正比例问题，可以使用单位法：先求一个单位的值，然后相乘以求出所需的值。对于 3 个苹果 90 便士，一个苹果的价格为 90 ÷ 3 = 30 便士。那么 7 个苹果花费 7 × 30 = 210 便士。

Proportion can also be represented in an equation: y = kx, where k is the constant of proportionality. In the apple example, cost = 30p per apple, so k = 30.

比例也可以用方程表示：y = kx，其中 k 是比例常数。在苹果的例子中，成本 = 每个苹果 30 便士，因此 k = 30。

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8. Inverse Proportion | 反比例

In inverse proportion, as one quantity increases, the other decreases in such a way that their product remains constant. For example, if it takes 4 workers 6 hours to paint a fence, then more workers will take less time. The total work done is constant, so if the number of workers doubles, the time halves.

在反比例中，当一个量增加时，另一个量减少，并且它们的乘积保持不变。例如，如果 4 个工人刷一道篱笆需要 6 小时，那么更多的工人所需时间更少。完成的总工作量是恒定的，因此如果工人数量翻倍，时间减半。

To solve inverse proportion problems, first find the constant product. In the example, total work = 4 workers × 6 hours = 24 worker-hours. If there are 8 workers, time = 24 ÷ 8 = 3 hours.

解决反比例问题，首先求出恒定的乘积。在该例中，总工作量 = 4 个工人 × 6 小时 = 24 个工人·小时。如果工人数为 8，时间 = 24 ÷ 8 = 3 小时。

The equation for inverse proportion is y = k / x, where k is the constant product. So xy = k.

反比例的方程为 y = k / x，其中 k 是恒定的乘积。因此 xy = k。

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9. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比之间的转换

Fluent conversion between these three forms is essential for solving ratio and proportion problems. Here are the key conversion methods:

• Fraction to decimal: divide the numerator by the denominator. e.g., 3/8 = 3 ÷ 8 = 0.375.
• Decimal to fraction: write the decimal as a fraction with a power of 10 as denominator and simplify. e.g., 0.24 = 24/100 = 6/25.
• Fraction to percentage: convert to a decimal first and multiply by 100, or find an equivalent fraction out of 100.
• Percentage to fraction: write over 100 and simplify.
• Decimal to percentage: multiply by 100 and add the % sign.
• Percentage to decimal: divide by 100.

熟练掌握这三种形式之间的转换对于解决比率和比例问题至关重要。以下是关键的转换方法：

• 分数化小数：用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。
• 小数化分数：将小数写成分母为10的幂的分数并化简。例如，0.24 = 24/100 = 6/25。
• 分数化百分比：先转换为小数再乘以100，或者找到一个分母为100的等值分数。
• 百分比化分数：写成分母为100的分数并化简。
• 小数化百分比：乘以100并加上%符号。
• 百分比化小数：除以100。

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10. Common Mistakes and How to Avoid Them | 常见错误及如何避免

Mistake 1: Forgetting to simplify ratios. Always check if all parts of a ratio can be divided by a common factor. If the ratio involves decimals or fractions, multiply by a common denominator to get whole numbers first.

错误1：忘记简化比率。始终检查比率的所有部分是否都可以被一个公因数除以。如果比率涉及小数或分数，先乘一个公分母以得到整数。

Mistake 2: Confusing percentage increase and decrease multipliers. Remember: for an increase of x%, the multiplier is (100 + x)/100. For a decrease of x%, it is (100 – x)/100. Do not use 0.x for an increase unless x is less than 100%.

错误2：混淆百分比增加和减少的乘数。记住：增加 x%，乘数为 (100 + x)/100。减少 x%，乘数为 (100 – x)/100。不要将增加时的 0.x 用于乘数，除非 x 小于 100%。

Mistake 3: In ratio sharing, forgetting to find the total number of parts first. Always add the parts, then divide the total quantity by the total parts to find one part.

错误3：在比率分配中，忘记先求总份数。始终将各部分相加，然后将总量除以总份数以得到一份的值。

Mistake 4: Using the wrong units in scale problems. Make sure that the units on both sides of the scale ratio match so that calculations are correct.

错误4：在比例尺问题中使用错误的单位。确保比例尺比率两边的单位匹配，以便计算正确。

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11. Real-world Applications | 实际应用

Percentages appear in finance: simple interest is calculated as a percentage of the principal. Ratios are used in recipes to adjust ingredient amounts. Proportions are used in science to understand relationships like speed = distance / time. A solid grasp of these topics allows students to tackle multi-step problems in exams and everyday life with logical reasoning.

百分比出现在金融中：单利是按照本金的百分比计算的。比率用于食谱中调整配料用量。比例用于科学中理解速度 = 距离/时间之类的关系。对这些主题的扎实掌握能让学生凭借逻辑推理，应对考试和日常生活中多步骤的问题。

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12. Summary and Practice Tips | 总结与练习建议

To master percentages, ratios and proportions, practice regularly with a variety of problems. Start by converting fluently between fractions, decimals and percentages. Then, work through word problems involving sharing ratios, scale maps, and percentage changes. Check your answers by working backwards, and always simplify ratios fully. Remember the key formulas for direct and inverse proportion, and practice identifying which type of proportion a situation describes.

要掌握百分比、比率和比例，需要定期练习各种问题。首先熟练地在分数、小数和百分比之间转换。然后，练习涉及比率分配、比例尺地图和百分比变化的应用题。通过逆向计算检查答案，并始终完全简化比率。记住正比例和反比例的关键公式，并练习判断情境所描述的是哪种比例类型。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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